# Thread: Lebesgue measure on sigma-fields of Borel sets

1. ## Lebesgue measure on sigma-fields of Borel sets

Hello, I am completely lost with this problem. Any guidance is greatly appreciated.

Let $\displaystyle \Omega=\{(x,y):0<x\leq 1,0<y\leq 1\}$, let $\displaystyle \mathcal{B}^2$ be the $\displaystyle \sigma$-field of Borel subsets of $\displaystyle \Omega$ and let $\displaystyle P=\lambda_2$ be the Lebesgue measure on $\displaystyle \mathcal{B}^2$. Let $\displaystyle A_0=\{(x,y):x>0,y>0,x+y\leq 1\}$.

a. Starting from the fact that $\displaystyle \mathcal{B}^2=\sigma(\mathcal{R}^{0,2})$, show that $\displaystyle A_0 \in \mathcal{B}^2$, where $\displaystyle \mathcal{R}^{0,2}$ is the collection of two-dimensional bounded rectangles of the form$\displaystyle B(a_1,b_1,a_2,b_2) = \{(x,y):a_1<x \leq b_1, a_2< y \leq b_2\}$ with $\displaystyle a_1,b_1,a_2,b_2 \in [0,1]$, and $\displaystyle a_1 \leq b_1, a_2 \leq b_2$

b. Let $\displaystyle P_*$ and $\displaystyle P^*$ be the inner and outer measures defined from$\displaystyle P=\lambda_2$. Prove that $\displaystyle P_*(A_0)=P^*(A_0)=P(A_0)=1/2$. (Hint: one way to do this is to construct sequences of sets $\displaystyle B_n$ and $\displaystyle C_n$ of known measure , e.g. unions of disjoint rectangles, with $\displaystyle B_n \downarrow A_0$ and $\displaystyle C_n \uparrow A_0$)

2. ## Re: Lebesgue measure on sigma-fields of Borel sets

Originally Posted by akolman
Hello, I am completely lost with this problem. Any guidance is greatly appreciated.

Let $\displaystyle \Omega=\{(x,y):0<x\leq 1,0<y\leq 1\}$, let $\displaystyle \mathcal{B}^2$ be the $\displaystyle \sigma$-field of Borel subsets of $\displaystyle \Omega$ and let $\displaystyle P=\lambda_2$ be the Lebesgue measure on $\displaystyle \mathcal{B}^2$. Let $\displaystyle A_0=\{(x,y):x>0,y>0,x+y\leq 1\}$.

a. Starting from the fact that $\displaystyle \mathcal{B}^2=\sigma(\mathcal{R}^{0,2})$, show that $\displaystyle A_0 \in \mathcal{B}^2$, where $\displaystyle \mathcal{R}^{0,2}$ is the collection of two-dimensional bounded rectangles of the form$\displaystyle B(a_1,b_1,a_2,b_2) = \{(x,y):a_1<x \leq b_1, a_2< y \leq b_2\}$ with $\displaystyle a_1,b_1,a_2,b_2 \in [0,1]$, and $\displaystyle a_1 \leq b_1, a_2 \leq b_2$
Each of those rectangles is borel....soo.

b. Let $\displaystyle P_*$ and $\displaystyle P^*$ be the inner and outer measures defined from$\displaystyle P=\lambda_2$. Prove that $\displaystyle P_*(A_0)=P^*(A_0)=P(A_0)=1/2$. (Hint: one way to do this is to construct sequences of sets $\displaystyle B_n$ and $\displaystyle C_n$ of known measure , e.g. unions of disjoint rectangles, with $\displaystyle B_n \downarrow A_0$ and $\displaystyle C_n \uparrow A_0$)

I mean, do you have any idea?

3. ## Re: Lebesgue measure on sigma-fields of Borel sets

I have an idea, but there are some gaps I'm not really sure how to fill in.

Let $\displaystyle B_n = \cup_{i=1}^N\{(x,y):\frac{i-1}{2^n} < x \leq \frac{i}{2^n}, 0 < y \leq 1- \frac{i-1}{2^n}\}$
By construction$\displaystyle B_{n+1} \subset B_n$ and$\displaystyle A_0 \subset \cap_{n=1}^\infty B_n.$
I am pretty sure that $\displaystyle \cap_{n=1}^\infty B_n \subset A_0$ but I don't know how to prove it.

For the second part, if the above is true $\displaystyle B_n \downarrow A_0$ so $\displaystyle P(B_n) \downarrow P(A_0)$ which should be 1/2. since $\displaystyle P=\lambda_2$and $\displaystyle \mathcal{B}^2$ is a $\displaystyle \sigma$ field the outer and inner measure should agree.

Is that on the right track? any hints to fill in the gaps? thanks again.